2.20. Explosive paths in the Samuelson overlapping-generations model (See Black, 1974; Brock, 1975; and Calvo, 1978a.) Consider the setup described in Problem 2.18. Assume that x is zero, and assume that utility is constant- relative-risk-aversion with <1 rather than logarithmic. Finally, assume for simplicity that n = 0.

(a) What is the behavior of an individual born at t as a function of Pr/Pt+1? Show that the amount of his or her endowment that the individual sells for money is an increasing function of P/P+1 and approaches zero as this ratio approaches zero.

(b) Suppose Po/P <1. How much of the good are the individuals born in period 0 planning to buy in period 1 from the individuals born then? What must P/P be for the individuals born in period 1 to want to supply this amount?

(c) Iterating this reasoning forward, what is the qualitative behavior of P/P+ over time? Does this represent an equilibrium path for the econ- omy?

(d) Can there be an equilibrium path with Po/P > 1?